3.351 \(\int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{F^{a+\frac{b}{(c+d x)^3}} \left (12 b^2 \log ^2(F) (c+d x)^6-4 b^3 \log ^3(F) (c+d x)^3+b^4 \log ^4(F)-24 b \log (F) (c+d x)^9+24 (c+d x)^{12}\right )}{3 b^5 d \log ^5(F) (c+d x)^{12}} \]

[Out]

-(F^(a + b/(c + d*x)^3)*(24*(c + d*x)^12 - 24*b*(c + d*x)^9*Log[F] + 12*b^2*(c + d*x)^6*Log[F]^2 - 4*b^3*(c +
d*x)^3*Log[F]^3 + b^4*Log[F]^4))/(3*b^5*d*(c + d*x)^12*Log[F]^5)

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Rubi [C]  time = 0.0440779, antiderivative size = 31, normalized size of antiderivative = 0.32, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac{F^a \text{Gamma}\left (5,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)/(c + d*x)^16,x]

[Out]

-(F^a*Gamma[5, -((b*Log[F])/(c + d*x)^3)])/(3*b^5*d*Log[F]^5)

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx &=-\frac{F^a \Gamma \left (5,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)}\\ \end{align*}

Mathematica [C]  time = 0.0086928, size = 31, normalized size = 0.32 \[ -\frac{F^a \text{Gamma}\left (5,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^16,x]

[Out]

-(F^a*Gamma[5, -((b*Log[F])/(c + d*x)^3)])/(3*b^5*d*Log[F]^5)

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Maple [B]  time = 0.29, size = 889, normalized size = 9.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x)

[Out]

(-d*c*(840*c^12-528*ln(F)*b*c^9+144*ln(F)^2*b^2*c^6-20*ln(F)^3*b^3*c^3+b^4*ln(F)^4)/b^5/ln(F)^5*x^2*exp((a+b/(
d*x+c)^3)*ln(F))+4*c^2*d^3*(-2730*c^9+990*ln(F)*b*c^6-126*ln(F)^2*b^2*c^3+5*ln(F)^3*b^3)/ln(F)^5/b^5*x^4*exp((
a+b/(d*x+c)^3)*ln(F))+8*c*d^4*(-3003*c^9+792*ln(F)*b*c^6-63*ln(F)^2*b^2*c^3+ln(F)^3*b^3)/ln(F)^5/b^5*x^5*exp((
a+b/(d*x+c)^3)*ln(F))-72*c^2*d^6*(715*c^6-88*ln(F)*b*c^3+2*ln(F)^2*b^2)/ln(F)^5/b^5*x^7*exp((a+b/(d*x+c)^3)*ln
(F))-36*c*d^7*(1430*c^6-110*ln(F)*b*c^3+ln(F)^2*b^2)/ln(F)^5/b^5*x^8*exp((a+b/(d*x+c)^3)*ln(F))+264*c^2*d^9*(-
91*c^3+2*b*ln(F))/ln(F)^5/b^5*x^10*exp((a+b/(d*x+c)^3)*ln(F))+24*c*d^10*(-455*c^3+4*b*ln(F))/ln(F)^5/b^5*x^11*
exp((a+b/(d*x+c)^3)*ln(F))-8*d^14/ln(F)^5/b^5*x^15*exp((a+b/(d*x+c)^3)*ln(F))-c^2*(120*c^12-96*ln(F)*b*c^9+36*
ln(F)^2*b^2*c^6-8*ln(F)^3*b^3*c^3+b^4*ln(F)^4)/b^5/ln(F)^5*x*exp((a+b/(d*x+c)^3)*ln(F))-1/3*d^2*(10920*c^12-52
80*ln(F)*b*c^9+1008*ln(F)^2*b^2*c^6-80*ln(F)^3*b^3*c^3+b^4*ln(F)^4)/ln(F)^5/b^5*x^3*exp((a+b/(d*x+c)^3)*ln(F))
+4/3*d^5*(-30030*c^9+5544*ln(F)*b*c^6-252*ln(F)^2*b^2*c^3+ln(F)^3*b^3)/ln(F)^5/b^5*x^6*exp((a+b/(d*x+c)^3)*ln(
F))-4*d^8*(10010*c^6-440*ln(F)*b*c^3+ln(F)^2*b^2)/ln(F)^5/b^5*x^9*exp((a+b/(d*x+c)^3)*ln(F))+8*d^11*(-455*c^3+
b*ln(F))/ln(F)^5/b^5*x^12*exp((a+b/(d*x+c)^3)*ln(F))-840*d^12*c^2/ln(F)^5/b^5*x^13*exp((a+b/(d*x+c)^3)*ln(F))-
120*d^13*c/ln(F)^5/b^5*x^14*exp((a+b/(d*x+c)^3)*ln(F))-1/3*(24*c^12-24*ln(F)*b*c^9+12*ln(F)^2*b^2*c^6-4*ln(F)^
3*b^3*c^3+b^4*ln(F)^4)*c^3/b^5/ln(F)^5/d*exp((a+b/(d*x+c)^3)*ln(F)))/(d*x+c)^15

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Maxima [B]  time = 1.11036, size = 1040, normalized size = 10.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="maxima")

[Out]

-1/3*(24*F^a*d^12*x^12 + 288*F^a*c*d^11*x^11 + 1584*F^a*c^2*d^10*x^10 + 24*F^a*c^12 - 24*F^a*b*c^9*log(F) + 12
*F^a*b^2*c^6*log(F)^2 + 24*(220*F^a*c^3*d^9 - F^a*b*d^9*log(F))*x^9 - 4*F^a*b^3*c^3*log(F)^3 + 216*(55*F^a*c^4
*d^8 - F^a*b*c*d^8*log(F))*x^8 + F^a*b^4*log(F)^4 + 864*(22*F^a*c^5*d^7 - F^a*b*c^2*d^7*log(F))*x^7 + 12*(1848
*F^a*c^6*d^6 - 168*F^a*b*c^3*d^6*log(F) + F^a*b^2*d^6*log(F)^2)*x^6 + 72*(264*F^a*c^7*d^5 - 42*F^a*b*c^4*d^5*l
og(F) + F^a*b^2*c*d^5*log(F)^2)*x^5 + 36*(330*F^a*c^8*d^4 - 84*F^a*b*c^5*d^4*log(F) + 5*F^a*b^2*c^2*d^4*log(F)
^2)*x^4 + 4*(1320*F^a*c^9*d^3 - 504*F^a*b*c^6*d^3*log(F) + 60*F^a*b^2*c^3*d^3*log(F)^2 - F^a*b^3*d^3*log(F)^3)
*x^3 + 12*(132*F^a*c^10*d^2 - 72*F^a*b*c^7*d^2*log(F) + 15*F^a*b^2*c^4*d^2*log(F)^2 - F^a*b^3*c*d^2*log(F)^3)*
x^2 + 12*(24*F^a*c^11*d - 18*F^a*b*c^8*d*log(F) + 6*F^a*b^2*c^5*d*log(F)^2 - F^a*b^3*c^2*d*log(F)^3)*x)*F^(b/(
d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(b^5*d^13*x^12*log(F)^5 + 12*b^5*c*d^12*x^11*log(F)^5 + 66*b^5*c^2*d
^11*x^10*log(F)^5 + 220*b^5*c^3*d^10*x^9*log(F)^5 + 495*b^5*c^4*d^9*x^8*log(F)^5 + 792*b^5*c^5*d^8*x^7*log(F)^
5 + 924*b^5*c^6*d^7*x^6*log(F)^5 + 792*b^5*c^7*d^6*x^5*log(F)^5 + 495*b^5*c^8*d^5*x^4*log(F)^5 + 220*b^5*c^9*d
^4*x^3*log(F)^5 + 66*b^5*c^10*d^3*x^2*log(F)^5 + 12*b^5*c^11*d^2*x*log(F)^5 + b^5*c^12*d*log(F)^5)

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Fricas [B]  time = 2.0205, size = 1381, normalized size = 14.39 \begin{align*} -\frac{{\left (24 \, d^{12} x^{12} + 288 \, c d^{11} x^{11} + 1584 \, c^{2} d^{10} x^{10} + 5280 \, c^{3} d^{9} x^{9} + 11880 \, c^{4} d^{8} x^{8} + 19008 \, c^{5} d^{7} x^{7} + 22176 \, c^{6} d^{6} x^{6} + 19008 \, c^{7} d^{5} x^{5} + 11880 \, c^{8} d^{4} x^{4} + 5280 \, c^{9} d^{3} x^{3} + 1584 \, c^{10} d^{2} x^{2} + 288 \, c^{11} d x + 24 \, c^{12} + b^{4} \log \left (F\right )^{4} - 4 \,{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (F\right )^{3} + 12 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} - 24 \,{\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \left (F\right )\right )} F^{\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \,{\left (b^{5} d^{13} x^{12} + 12 \, b^{5} c d^{12} x^{11} + 66 \, b^{5} c^{2} d^{11} x^{10} + 220 \, b^{5} c^{3} d^{10} x^{9} + 495 \, b^{5} c^{4} d^{9} x^{8} + 792 \, b^{5} c^{5} d^{8} x^{7} + 924 \, b^{5} c^{6} d^{7} x^{6} + 792 \, b^{5} c^{7} d^{6} x^{5} + 495 \, b^{5} c^{8} d^{5} x^{4} + 220 \, b^{5} c^{9} d^{4} x^{3} + 66 \, b^{5} c^{10} d^{3} x^{2} + 12 \, b^{5} c^{11} d^{2} x + b^{5} c^{12} d\right )} \log \left (F\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="fricas")

[Out]

-1/3*(24*d^12*x^12 + 288*c*d^11*x^11 + 1584*c^2*d^10*x^10 + 5280*c^3*d^9*x^9 + 11880*c^4*d^8*x^8 + 19008*c^5*d
^7*x^7 + 22176*c^6*d^6*x^6 + 19008*c^7*d^5*x^5 + 11880*c^8*d^4*x^4 + 5280*c^9*d^3*x^3 + 1584*c^10*d^2*x^2 + 28
8*c^11*d*x + 24*c^12 + b^4*log(F)^4 - 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(F)^3 + 1
2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4*d^2*x^2 + 6*b^2*c^5*d*
x + b^2*c^6)*log(F)^2 - 24*(b*d^9*x^9 + 9*b*c*d^8*x^8 + 36*b*c^2*d^7*x^7 + 84*b*c^3*d^6*x^6 + 126*b*c^4*d^5*x^
5 + 126*b*c^5*d^4*x^4 + 84*b*c^6*d^3*x^3 + 36*b*c^7*d^2*x^2 + 9*b*c^8*d*x + b*c^9)*log(F))*F^((a*d^3*x^3 + 3*a
*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/((b^5*d^13*x^12 + 12*b^5*c*d^
12*x^11 + 66*b^5*c^2*d^11*x^10 + 220*b^5*c^3*d^10*x^9 + 495*b^5*c^4*d^9*x^8 + 792*b^5*c^5*d^8*x^7 + 924*b^5*c^
6*d^7*x^6 + 792*b^5*c^7*d^6*x^5 + 495*b^5*c^8*d^5*x^4 + 220*b^5*c^9*d^4*x^3 + 66*b^5*c^10*d^3*x^2 + 12*b^5*c^1
1*d^2*x + b^5*c^12*d)*log(F)^5)

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Sympy [B]  time = 1.06645, size = 760, normalized size = 7.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**3)/(d*x+c)**16,x)

[Out]

F**(a + b/(c + d*x)**3)*(-b**4*log(F)**4 + 4*b**3*c**3*log(F)**3 + 12*b**3*c**2*d*x*log(F)**3 + 12*b**3*c*d**2
*x**2*log(F)**3 + 4*b**3*d**3*x**3*log(F)**3 - 12*b**2*c**6*log(F)**2 - 72*b**2*c**5*d*x*log(F)**2 - 180*b**2*
c**4*d**2*x**2*log(F)**2 - 240*b**2*c**3*d**3*x**3*log(F)**2 - 180*b**2*c**2*d**4*x**4*log(F)**2 - 72*b**2*c*d
**5*x**5*log(F)**2 - 12*b**2*d**6*x**6*log(F)**2 + 24*b*c**9*log(F) + 216*b*c**8*d*x*log(F) + 864*b*c**7*d**2*
x**2*log(F) + 2016*b*c**6*d**3*x**3*log(F) + 3024*b*c**5*d**4*x**4*log(F) + 3024*b*c**4*d**5*x**5*log(F) + 201
6*b*c**3*d**6*x**6*log(F) + 864*b*c**2*d**7*x**7*log(F) + 216*b*c*d**8*x**8*log(F) + 24*b*d**9*x**9*log(F) - 2
4*c**12 - 288*c**11*d*x - 1584*c**10*d**2*x**2 - 5280*c**9*d**3*x**3 - 11880*c**8*d**4*x**4 - 19008*c**7*d**5*
x**5 - 22176*c**6*d**6*x**6 - 19008*c**5*d**7*x**7 - 11880*c**4*d**8*x**8 - 5280*c**3*d**9*x**9 - 1584*c**2*d*
*10*x**10 - 288*c*d**11*x**11 - 24*d**12*x**12)/(3*b**5*c**12*d*log(F)**5 + 36*b**5*c**11*d**2*x*log(F)**5 + 1
98*b**5*c**10*d**3*x**2*log(F)**5 + 660*b**5*c**9*d**4*x**3*log(F)**5 + 1485*b**5*c**8*d**5*x**4*log(F)**5 + 2
376*b**5*c**7*d**6*x**5*log(F)**5 + 2772*b**5*c**6*d**7*x**6*log(F)**5 + 2376*b**5*c**5*d**8*x**7*log(F)**5 +
1485*b**5*c**4*d**9*x**8*log(F)**5 + 660*b**5*c**3*d**10*x**9*log(F)**5 + 198*b**5*c**2*d**11*x**10*log(F)**5
+ 36*b**5*c*d**12*x**11*log(F)**5 + 3*b**5*d**13*x**12*log(F)**5)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{16}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^16, x)